# Potentials of the Heun class: The triconfluent case

@article{Batic2015PotentialsOT, title={Potentials of the Heun class: The triconfluent case}, author={David Batic and D. Mills-Howell and Marek Nowakowski}, journal={Journal of Mathematical Physics}, year={2015}, volume={56}, pages={052106} }

We study special classes of potentials for which the one-dimensional (or radial) Schrodinger equation can be reduced to a triconfluent Heun equation by a suitable coordinate transformation together with an additional transformation of the wave function. In particular, we analyze the behaviour of those subclasses of the potential arising when the ordinary differential equation governing the coordinate transformation admits explicit analytic solutions in terms of the radial variable. Furthermore…

## 16 Citations

Hermite function solutions of the Schrödinger equation for the sextic oscillator

- Physics, MathematicsPhysica Scripta
- 2020

We examine the conditions under which the solution of the radial stationary Schrödinger equation for the sextic anharmonic oscillator can be expanded in terms of Hermite functions. We find that this…

Semicommuting and Commuting Operators for the Heun Family

- Mathematics
- 2017

We derive the most general families of first- and second-order differential operators semicommuting with the Heun class differential operators. Among these families, we classify all the families that…

A Conditionally Integrable Bi-confluent Heun Potential Involving Inverse Square Root and Centrifugal Barrier Terms

- Mathematics
- 2017

Abstract We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions.…

Discretization of Natanzon potentials

- Mathematics
- 2016

Abstract.We show that the Natanzon family of potentials is necessarily dropped into a restricted set of distinct potentials involving a fewer number of independent parameters if the potential term in…

Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

- MathematicsAdvances in High Energy Physics
- 2018

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from…

F eb 2 01 9 Scalar scattering in Reissner-Nordström spacetime

- Mathematics
- 2019

In this paper, we solve the scattering phase shift of a scalar field scattering in Reissner-Nordström spacetime. The scattering boundary condition is determined by analyzing the asymptotic solution…

## References

SHOWING 1-10 OF 34 REFERENCES

Potentials of the Heun class

- Mathematics
- 2013

We review different methods of generating potentials such that the one-dimensional Schrödinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with…

Study of the one-dimensional Schroedinger equation generated from the hypergeometric equation

- Mathematics, Physics
- 1999

Original English Summary. - A systematic method of constructing potentials, for which the one-variable Schroedinger equation can be solved in terms of the hypergeometric (HGM) function, is presented.…

The Schrödinger equation

- Mathematics
- 1998

By making use of an ansatz for the eigenfunction, we obtain the exact solutions to the Schrödinger equation with the anharmonic potential, V (r) = ar 2 + br−4 + cr−6, both in three dimensions and in…

High Energy Eigenfunctions of One-Dimensional Schrödinger Operators with Polynomial Potentials

- Mathematics
- 2007

For a class of one-dimensional Schrödinger operators with polynomial potentials that includes Hermitian and PT-symmetric operators, we show that the zeros of scaled eigenfunctions have a limit…

Liouville Transformation and Exactly Solvable Schrodinger Equations

- Physics, Mathematics
- 1997

The present paper discusses the connectionbetween exactly solvable Schrodinger equations and theLiouville transformation. This transformation yields alarge class of exactly solvable potentials,…

Anharmonic Oscillator Equations:Treatment Parallel to Mathieu Equation

- Mathematics
- 2004

The treatment of anharmonic oscillators (including double-wells) by instanton methods is wellknown. The alternative differential equation method is not so wellknown. Here we reformulate the latter…

Effective-field-theory model for the fractional quantum Hall effect.

- PhysicsPhysical review letters
- 1989

A field-theory model for the fractional quantum Hall effect and an approximate coarse-grained version of the same model are derived, and a Landau-Ginzburg theory similar to that of Girvin is constructed.

Quasi-Exactly Solvable Models in Quantum Mechanics

- Mathematics, Physics
- 1994

QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN? Introduction Completely algebraizable spectral problems The quartic oscillator The sextic oscillator Non-perturbative effects in an explicit form and…

ZEROS OF EIGENFUNCTIONS OF SOME ANHARMONIC OSCILLATORS

- Mathematics
- 2006

We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions…

A class of solvable potentials

- Physics
- 1962

SummaryA systematic method of constructing (velocity-independent) potentials, for which thes-wave Schrödinger equation can be solved in terms of known functions, is presented. Several such examples…